A general fractal model of flow and solute transport in randomly heterogeneous porous media

Chen, Kuan‐Chih; Hsu, Kuo Chin

doi:10.1029/2007wr005934

Cited by 15 publications

(17 citation statements)

References 36 publications

(62 reference statements)

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“…Observed multiscale behaviors of several subsurface fluid flow and transport variables in diverse geologic settings, at various sites and across a wide range of sampling domain scales have been found to be mutually consistent with a view of log permeability as a statistically isotropic or anisotropic self‐affine random field characterized by an α ‐order power variogram (semi‐structure function of order 2 when α = 2, width function when α < 2) γ α ( s ) = C 0 s αH (Neuman and Di Federico, ; Chen and Hsu, ; Neuman, ) where s is a scalar measure of distance (lag, defined in equivalent isotropic coordinates when the field is statistically anisotropic), C 0 being a constant, 0 < α ≤ 2 a Lévy index (Samorodnitsky and Taqqu, ) and 0 < H < 1 a Hurst exponent (Mandelbrot and Van Ness, ). The α ‐order power variogram scales as γ α ( rs ) = r αH γ α ( s ) for any r > 0.…”

Section: Introductionmentioning

confidence: 64%

Numerical investigation of apparent multifractality of samples from processes subordinated to truncated fBm

Guadagnini

Neuman

Riva

2011

Hydrological Processes

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Abstract:We investigate numerically apparent multi-fractal behavior of samples from synthetically generated processes subordinated to truncated fractional Brownian motion (tfBm) on finite domains. We are motivated by the recognition that many earth and environmental (including hydrological) variables appear to be self-affine (monofractal) or multifractal with Gaussian or heavy-tailed distributions. The literature considers self-affine and multifractal types of scaling to be fundamentally different, the first arising from additive and the second from multiplicative random fields or processes. It has been demonstrated theoretically by one of us that square or absolute increments of samples from Gaussian/Lévy processes subordinated to tfBm exhibit apparent/ spurious multifractality at intermediate ranges of separation lags, with breakdown in power-law scaling at small and large lags as is commonly exhibited by real data. A preliminary numerical demonstration of apparent multifractality by the same author was limited to Gaussian fields having nearest neighbor autocorrelations and led to rather noisy results. Here, we adopt a new generation scheme that allows us to investigate apparent multifractal behaviors of samples taken from a broad range of processes including Gaussian with and without symmetric Lévy and log-normal (as well as potentially other) subordinators. Our results shed new light on the nature of apparent multifractality, which has wide implications vis-a-vis the scaling of many hydrological as well as other earth and environmental variables.

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Section: Introductionmentioning

confidence: 64%

Numerical investigation of apparent multifractality of samples from processes subordinated to truncated fBm

Guadagnini

Neuman

Riva

2011

Hydrological Processes

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“…10 and 13 can be found in Chen (2007). If both a lower cutoff n l and an upper cutoff n u are considered, the truncated variogram becomes (Di Federico et al 1999):…”

Section: Two-point Statistics Of the Fractal Random Fieldmentioning

confidence: 99%

Multiscale flow and transport model in three-dimensional fractal porous media

Hsu

Chen

2010

Stoch Environ Res Risk Assess

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This paper proposes a multiscale flow and transport model which can be used in three-dimensional fractal random fields. The fractal random field effectively describes a field with a high degree of variability to satisfy the one-point statistics of Levy-stable distribution and the two-point statistics of fractional Levy motion (fLm). To overcome the difficulty of using infinite variance of Levystable distribution and to provide the physical meaning of a finite domain in real space, truncated power variograms are utilized for the fLm fields. The fLm model is general in the sense that both stationary and commonly used fractional Brownian motion (fBm) models are its special cases. When the upper cutoff of the truncated power variogram is close to the lower cutoff, the stationary model is well approximated. The commonly used fBm model is recovered when the Levy index of fLm is 2. Flow and solute transport were analyzed using the first-order perturbation method. Mean velocity, velocity covariance, and effective hydraulic conductivity in a three-dimensional fractal random field were derived. Analytical results for particle displacement covariance and macrodispersion coefficients are also presented. The results show that the plume in an fLm field moves slower at early time and has more significant longtailing behavior at late time than in fBm or stationary exponential fields. The proposed fractal transport model has broader applications than those of stationary and fBm models. Flow and solute transport can be simulated for various scenarios by adjusting the Levy index and cutoffs of fLm to yield more accurate modeling results.

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“…This was taken as an indication that the characteristic lengths of the modeled drainage areas are independently self‐affine in the one‐dimensional space Z of total channel length, rendering the network topologically anisotropic. Being self‐affine means forming a random function of the deterministic length measure Z (or, equivalently, a random field in Z ) and possessing a power (semi)variogram (or width function) γ () = A 0 αH where is a scalar measure of distance (lag) along the Z axis, α being a Levy index [ Chen and Hsu , 2007]. The power variogram scales as γ ( r ) = r αH γ () for any r > 0.…”

Section: Introductionmentioning

confidence: 99%

Hortonian scaling and effect of cutoffs on statistics of self‐affine river networks

Neuman

2009

Water Resources Research

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[1] The planar structure of river networks exhibits fractal properties. In particular, available data indicate a tendency for drainage areas A to be distributed according to a power law; their boundaries and main channels to form self-affine curves; the characteristic lengths of drainage areas to be independently self-affine in the onedimensional space Z of total channel length, rendering the network elongated (and anisotropic); the inverse network density A/Z to be either constant or self-affine in Z, depending on whether or not the network fills the two-dimensional space of A; and (as found in a recent study) areas of given size, vegetative cover, and mean steady state soil moisture, weighted by their flow distance from the basin outlet, to be self-affine in the one-dimensional space of this distance. Various theoretical and semiempirical relationships have been proposed among exponents defining some of these and other scale dependencies. Expressions have been proposed for ensemble moments of A and some length measures associated with finite size river basins. We present a new model that views any self-affine basin property, Y(X), as belonging to an infinite hierarchy of mutually uncorrelated, statistically homogeneous random functions defined on elementary subbasins, each of which is characterized by a unique integral (spatial correlation) scale l. We cite a mathematically rigorous hydrologic rationale for our model and use it in conjunction with published scaling relations to obtain the probability density function of l; to derive analytical expressions for ensemble analogs of Horton's scaling laws; to deduce from them that streams of any Horton-Strahler order w are associated with integral scales l w l l w+1 , where the ratio l w+1 /l w is a constant independent of w; to develop analytical expressions relating statistical moments of Y(X) to arbitrary lower and upper cutoff scales that may (but need not) be taken to represent data support and maximum watershed size; to describe ways of estimating the corresponding parameters; and to provide a theoretical basis for the heretofore unexplained observation that self-affine amplitude fluctuations of basin boundaries and main channels (as measured by their variance), having a common Hurst exponent, are larger in the former than in the latter.

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A general fractal model of flow and solute transport in randomly heterogeneous porous media

Cited by 15 publications

References 36 publications

Numerical investigation of apparent multifractality of samples from processes subordinated to truncated fBm

Numerical investigation of apparent multifractality of samples from processes subordinated to truncated fBm

Multiscale flow and transport model in three-dimensional fractal porous media

Hortonian scaling and effect of cutoffs on statistics of self‐affine river networks

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